First-Order Theorem Proving and Vampire

... In the TPTP Syntax The TPTP library (Thousands of Problems for Theorem Provers), http://www.tptp.org contains a large collection of first-order problems. For representing these problems it uses the TPTP syntax, which is understood by all modern theorem provers, including Vampire. In the TPTP syntax ...

... In the TPTP Syntax The TPTP library (Thousands of Problems for Theorem Provers), http://www.tptp.org contains a large collection of first-order problems. For representing these problems it uses the TPTP syntax, which is understood by all modern theorem provers, including Vampire. In the TPTP syntax ...

a semantic perspective - Institute for Logic, Language and

... There is no single answer. For example, modal logic can be given an algebraic semantics, and under this interpretation modal logic is a tool for talking about what are known as boolean algebras with operators. And modal logic can be given a topological semantics, so it can also be viewed as a tool f ...

... There is no single answer. For example, modal logic can be given an algebraic semantics, and under this interpretation modal logic is a tool for talking about what are known as boolean algebras with operators. And modal logic can be given a topological semantics, so it can also be viewed as a tool f ...

Model Theory of Modal Logic, Chapter in: Handbook of Modal Logic

... between the (ﬁrst-order) Kripke structure semantics and the (second-order) frame semantics, give rise to very distinct model theoretic ﬂavours, each with their own tradition in the model theory of modal logic. Still, these two semantics meet through the notion of a general frame (closely related to ...

... between the (ﬁrst-order) Kripke structure semantics and the (second-order) frame semantics, give rise to very distinct model theoretic ﬂavours, each with their own tradition in the model theory of modal logic. Still, these two semantics meet through the notion of a general frame (closely related to ...

Structural Proof Theory

... structural proof theory belongs, with a few exceptions, to what can be described as computational proof theory. Since 1970, a branch of proof theory known as constructive type theory has been developed. A theorem typically states that a certain claim holds under given assumptions. The basic idea of ...

... structural proof theory belongs, with a few exceptions, to what can be described as computational proof theory. Since 1970, a branch of proof theory known as constructive type theory has been developed. A theorem typically states that a certain claim holds under given assumptions. The basic idea of ...

Labeled Natural Deduction for Temporal Logics

... Table 1.1 presents a, clearly not comprehensive, map of temporal logics, which will help clarify the main contributions of this thesis. The first column presents logics whose underlying flow of time is linear, while in the second and third column we have branching logics, i.e., the flow of time is a ...

... Table 1.1 presents a, clearly not comprehensive, map of temporal logics, which will help clarify the main contributions of this thesis. The first column presents logics whose underlying flow of time is linear, while in the second and third column we have branching logics, i.e., the flow of time is a ...

Ribbon Proofs - A Proof System for the Logic of Bunched Implications

... BI is presented in [32] and elsewhere as a logic for reasoning about resource. With its two conjunctions it can contrast the notion that A and B share resources (A ∧ B) and the notion that they have disjoint resources (A ∗ B). We are particularly interested in the work of Ishtiaq, O’Hearn, Reynolds, ...

... BI is presented in [32] and elsewhere as a logic for reasoning about resource. With its two conjunctions it can contrast the notion that A and B share resources (A ∧ B) and the notion that they have disjoint resources (A ∗ B). We are particularly interested in the work of Ishtiaq, O’Hearn, Reynolds, ...

Logic Part II: Intuitionistic Logic and Natural Deduction

... 2. This proof contains of a proof of a. 3. It also contains a proof of b . 4. So if we take the proof of b and put it together with the proof of a, we obtain a proof of b ...

... 2. This proof contains of a proof of a. 3. It also contains a proof of b . 4. So if we take the proof of b and put it together with the proof of a, we obtain a proof of b ...

Goal-directed Proof Theory

... S and check by the theorem prover whether ; ^ S ! A. Again, no matter how it is ecient our theorem-prover, it is obvious that there are better methods of performing this task, for instance one attempts a proof of A from ; and determine as far as the proof proceeds what should be added to ; (i.e. a ...

... S and check by the theorem prover whether ; ^ S ! A. Again, no matter how it is ecient our theorem-prover, it is obvious that there are better methods of performing this task, for instance one attempts a proof of A from ; and determine as far as the proof proceeds what should be added to ; (i.e. a ...

SEQUENT SYSTEMS FOR MODAL LOGICS

... The familiar cut-free sequent calculus for monadic predicate logic can serve as a starting point for defining a cut-free ordinary sequent system for S5 with side-conditions on the introduction rules for 2 on the right and 3 on the left of the sequent arrow. The side conditions are simple, though the ...

... The familiar cut-free sequent calculus for monadic predicate logic can serve as a starting point for defining a cut-free ordinary sequent system for S5 with side-conditions on the introduction rules for 2 on the right and 3 on the left of the sequent arrow. The side conditions are simple, though the ...

Higher Order Logic - Theory and Logic Group

... try to present a survey of some issues and results, without any pretense of completeness. Our choice of topics is driven by an attempt to cover the foundational aspects of higher order logic, and also to brie y point to some areas of current and potential applications. The chapter falls into two par ...

... try to present a survey of some issues and results, without any pretense of completeness. Our choice of topics is driven by an attempt to cover the foundational aspects of higher order logic, and also to brie y point to some areas of current and potential applications. The chapter falls into two par ...

Higher Order Logic - Indiana University

... try to present a survey of some issues and results, without any pretense of completeness. Our choice of topics is driven by an attempt to cover the foundational aspects of higher order logic, and also to brie y point to some areas of current and potential applications. The chapter falls into two par ...

... try to present a survey of some issues and results, without any pretense of completeness. Our choice of topics is driven by an attempt to cover the foundational aspects of higher order logic, and also to brie y point to some areas of current and potential applications. The chapter falls into two par ...

Full Text - Institute for Logic, Language and Computation

... serious analysis of the conditional. Their work contains at the same time philosophy, but more importantly, rigorous syntax and semantics considerations which end up in completeness and decidability proofs for the systems that they present. But for each of these systems, counterexamples were present ...

... serious analysis of the conditional. Their work contains at the same time philosophy, but more importantly, rigorous syntax and semantics considerations which end up in completeness and decidability proofs for the systems that they present. But for each of these systems, counterexamples were present ...

Logic in Nonmonotonic Reasoning

... discuss what ought to be deduced under ordinary circumstances. Minsky was also one of the first who pointed to monotonicity of logical systems as a source of the problem: Monotonicity: ... In any logistic system, all the axioms are necessarily “permissive” - they all help to permit new inferences to ...

... discuss what ought to be deduced under ordinary circumstances. Minsky was also one of the first who pointed to monotonicity of logical systems as a source of the problem: Monotonicity: ... In any logistic system, all the axioms are necessarily “permissive” - they all help to permit new inferences to ...

Proofs in Propositional Logic

... This tactic can be useful for avoiding proof duplication inside some interactive proof. Notice that the scope of the declaration H :B is limited to the second subgoal. If a proof of B is needed elsewhere, it would be better to prove a lemma stating B. Remark : Sometimes the overuse of assert may lea ...

... This tactic can be useful for avoiding proof duplication inside some interactive proof. Notice that the scope of the declaration H :B is limited to the second subgoal. If a proof of B is needed elsewhere, it would be better to prove a lemma stating B. Remark : Sometimes the overuse of assert may lea ...

... This tactic can be useful for avoiding proof duplication inside some interactive proof. Notice that the scope of the declaration H :B is limited to the second subgoal. If a proof of B is needed elsewhere, it would be better to prove a lemma stating B. Remark : Sometimes the overuse of assert may lea ...

Deductive Databases with Universally Quantified Conditions

... a n-ary predicate symbol, t1, ..., tn are terms exactly one of are either constant or variable symbols. a n-ary predicate symbol, t1, ..., tn are terms exactly one of are either constant or variable symbols. ...

... a n-ary predicate symbol, t1, ..., tn are terms exactly one of are either constant or variable symbols. a n-ary predicate symbol, t1, ..., tn are terms exactly one of are either constant or variable symbols. ...

Cut-elimination for provability logics and some results in display logic

... of the cut-rule from a derivation in the sequent calculus thus leading to a cutfree derivation. This is a central result in the proof-theory of a logic. In 1983, Valentini [71] presented a syntactic proof of cut-elimination for the provability logic GL for a traditional Gentzen sequent calculus buil ...

... of the cut-rule from a derivation in the sequent calculus thus leading to a cutfree derivation. This is a central result in the proof-theory of a logic. In 1983, Valentini [71] presented a syntactic proof of cut-elimination for the provability logic GL for a traditional Gentzen sequent calculus buil ...

Handling Exceptions in nonmonotonic reasoning

... Our proceeding to determine the theorems of a defeasible axiomatic basis is, then, to impose some conditions upon candidates to be considered an expansion. The theory associated with an expansion is what we call an admissible set of theorems of a defeasible axiomatic basis, this is our equivalent no ...

... Our proceeding to determine the theorems of a defeasible axiomatic basis is, then, to impose some conditions upon candidates to be considered an expansion. The theory associated with an expansion is what we call an admissible set of theorems of a defeasible axiomatic basis, this is our equivalent no ...

The Foundations

... Problem: when will you say that the sentence: It_is_raining is true ? => The proposition:” It_is_raining” is true iff the condition (or fact) that the sentence is intended to state really occurs(happens, exists) in the situation which the proposition is intended to describe. =>Example: Since it is ...

... Problem: when will you say that the sentence: It_is_raining is true ? => The proposition:” It_is_raining” is true iff the condition (or fact) that the sentence is intended to state really occurs(happens, exists) in the situation which the proposition is intended to describe. =>Example: Since it is ...

Formal deduction in propositional logic

... ’Contrariwise,’ continued Tweedledee, ’if it was so, it might be; and if it were so, it would be; but as it isn’t, it ain’t. That’s logic.’ (Lewis Caroll, “Alice in Wonderland”) Formal deduction in propositional logic ...

... ’Contrariwise,’ continued Tweedledee, ’if it was so, it might be; and if it were so, it would be; but as it isn’t, it ain’t. That’s logic.’ (Lewis Caroll, “Alice in Wonderland”) Formal deduction in propositional logic ...

In order to define the notion of proof rigorously, we would have to

... discharging mechanism but they all involve some form of tagging (with “new” variable). For example, the rule formalizing the process that we have just described to prove an implication, A ⇒ B, known as ⇒-introduction, uses a tagging mechanism described precisely in Definition 1.2.1. Now, the rule th ...

... discharging mechanism but they all involve some form of tagging (with “new” variable). For example, the rule formalizing the process that we have just described to prove an implication, A ⇒ B, known as ⇒-introduction, uses a tagging mechanism described precisely in Definition 1.2.1. Now, the rule th ...

Modal Logic - Web Services Overview

... Modal Logic: Semantics • Semantics is given in terms of Kripke Structures (also known as possible worlds structures) • Due to American logician Saul Kripke, City University of NY • A Kripke Structure is (W, R) – W is a set of possible worlds – R : W W is an binary accessibility relation over W – ...

... Modal Logic: Semantics • Semantics is given in terms of Kripke Structures (also known as possible worlds structures) • Due to American logician Saul Kripke, City University of NY • A Kripke Structure is (W, R) – W is a set of possible worlds – R : W W is an binary accessibility relation over W – ...

Combinaison des logiques temporelle et déontique pour la

... stay very pleasant. I learnt a lot about deontic logics and action logics. The collaboration that followed was really stimulating until the end of my PhD, and the logic proposed in this dissertation is its fruit. My acknowledgment then goes to Philippe Balbiani, for all the discussions we had, about ...

... stay very pleasant. I learnt a lot about deontic logics and action logics. The collaboration that followed was really stimulating until the end of my PhD, and the logic proposed in this dissertation is its fruit. My acknowledgment then goes to Philippe Balbiani, for all the discussions we had, about ...

LPF and MPLω — A Logical Comparison of VDM SL and COLD-K

... For LPF, the underlying notion of logical consequence is the third one of the abovementioned notions. The first notion underlies Owe’s weak logic [Owe84]. The second notion underlies the logic PFOL which is presented in [GL90] together with reflections on the semantic options for three-valued logics ...

... For LPF, the underlying notion of logical consequence is the third one of the abovementioned notions. The first notion underlies Owe’s weak logic [Owe84]. The second notion underlies the logic PFOL which is presented in [GL90] together with reflections on the semantic options for three-valued logics ...

The Foundations

... (or fact) that the proposition is intended to represent occurs (happens, exists) in the situation which the proposition is intended to describe. =>Example: Since it is not raining now (the current situation), the statement “It_is_raining” is false (in the current situation). But if it were raining n ...

... (or fact) that the proposition is intended to represent occurs (happens, exists) in the situation which the proposition is intended to describe. =>Example: Since it is not raining now (the current situation), the statement “It_is_raining” is false (in the current situation). But if it were raining n ...